Divide using synthetic division $$ ( -2x^{4}+3x^{3}-2x^{2}+3 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&-2&3&-2&0&3\\& & 6& -27& 87& \color{black}{-261} \\ \hline &\color{blue}{-2}&\color{blue}{9}&\color{blue}{-29}&\color{blue}{87}&\color{orangered}{-258} \end{array} $$The solution is:
$$ \frac{ -2x^{4}+3x^{3}-2x^{2}+3 }{ x+3 } = \color{blue}{-2x^{3}+9x^{2}-29x+87} \color{red}{~-~} \frac{ \color{red}{ 258 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&-2&3&-2&0&3\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ -2 }&3&-2&0&3\\& & & & & \\ \hline &\color{orangered}{-2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&-2&3&-2&0&3\\& & \color{blue}{6} & & & \\ \hline &\color{blue}{-2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 6 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}-3&-2&\color{orangered}{ 3 }&-2&0&3\\& & \color{orangered}{6} & & & \\ \hline &-2&\color{orangered}{9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 9 } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&-2&3&-2&0&3\\& & 6& \color{blue}{-27} & & \\ \hline &-2&\color{blue}{9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -29 } $
$$ \begin{array}{c|rrrrr}-3&-2&3&\color{orangered}{ -2 }&0&3\\& & 6& \color{orangered}{-27} & & \\ \hline &-2&9&\color{orangered}{-29}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -29 \right) } = \color{blue}{ 87 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&-2&3&-2&0&3\\& & 6& -27& \color{blue}{87} & \\ \hline &-2&9&\color{blue}{-29}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 87 } = \color{orangered}{ 87 } $
$$ \begin{array}{c|rrrrr}-3&-2&3&-2&\color{orangered}{ 0 }&3\\& & 6& -27& \color{orangered}{87} & \\ \hline &-2&9&-29&\color{orangered}{87}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 87 } = \color{blue}{ -261 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&-2&3&-2&0&3\\& & 6& -27& 87& \color{blue}{-261} \\ \hline &-2&9&-29&\color{blue}{87}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -261 \right) } = \color{orangered}{ -258 } $
$$ \begin{array}{c|rrrrr}-3&-2&3&-2&0&\color{orangered}{ 3 }\\& & 6& -27& 87& \color{orangered}{-261} \\ \hline &\color{blue}{-2}&\color{blue}{9}&\color{blue}{-29}&\color{blue}{87}&\color{orangered}{-258} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{3}+9x^{2}-29x+87 } $ with a remainder of $ \color{red}{ -258 } $.